Aver is a formal word that comes from the Latin root verus, meaning "true." In the courtroom, aver means to officially state the truth, but you can use aver anytime you need to confirm that something is a fact, like so:
In a Monday court filing, they said they "deny that they engaged in any wrongful, illegal or improper conduct, and aver that each of them is innocent." (Reuters)To avow is to announce something to everyone. Avow is often used in the adjective form: If you are an avowed cat lover, everyone knows you love cats. Here are some examples:As a photographer, I can aver that Post photographer Jabin Botsford is a bona fide graphic genius when it comes to covering the White House beat. (Washington Post)
That effort, they avowed, will "send a strong signal to the Chinese Communist Party that the American people are committed to defending U.S. interests in the Indo-Pacific." (Salon)
A first-time candidate and an avowed science-fiction fan, Mr. Jones likened the race to “a story out of a sci-fi novel,” he said. (New York Times)
So remember, to aver is to verify, and to avow is to tell the whole world.